After exporting my character as an FBX, with the goal of designing and animating within Maya using the included rig and facial expression blendshapes, there is a weird bug with the head/face part of the geometry. The local, relative positions of all the vertices and faces snap to a location elsewhere on the head. While it looks fine and renders fine, this makes it impossible to rig constraints onto the 'skin' of the character (to track objects on the face geometry) or to use the Make Live/Make Paintable features in Maya (for texture painting and hair/beard designing).
To recreate the problem, import a character into Maya, create a simple object like a poly sphere, and try to constrain it to the face using 'Rivet' or 'Point to Poly' which requires you to select a vertex on the character. The constraint will automatically snap to a random place inside or across the head somewhere.
Note: this error ONLY occurs on the head/face geometry. The constraints and paint features seem to work fine on other parts of the body. For example, if you use Paint FX/Make Paintable and draw a stroke across the chest, it works as expected. But when you draw the brush across the clavicle into the neck area, the brush origin snaps automatically to a random point on the head.